The diagonal Ising susceptibility

Abstract

We use the recently derived form factor expansions of the diagonal two-point correlation function of the square Ising model to study the susceptibility for a magnetic field applied only to one diagonal of the lattice, for the isotropic Ising model. We exactly evaluate the one and two particle contributions ${\chi }_d^{(1)}$ and ${\chi }_d^{(2)}$ of the corresponding susceptibility, and obtain linear differential equations for the three and four particle contributions, as well as the five particle contribution ${\chi }_d^{(5)}(t)$, but only modulo a given prime. We use these exact linear differential equations to show that, not only the russian-doll structure, but also the direct sum structure on the linear differential operators for the $n$-particle contributions ${\chi }_d^{(n)}$ are quite directly inherited from the direct sum structure on the form factors $f^{(n)}$. We show that the $n^{th}$ particle contributions ${\chi }_d^{(n)}$ have their singularities at roots of unity. These singularities become dense on the unit circle $|\sinh 2E_v/kT\sinh 2E_h/kT|=1$ as $n\to \infty$.

18 pages

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Cited by in corpus (22)

• New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain
• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• Asymptotic analysis and quantum integrable models
• High order Fuchsian equations for the square lattice Ising model: ${\tilde{\chi }}^{(5)}$
• The importance of the Ising model
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Natural Boundary for a Sum Involving Toeplitz Determinants
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
• Modular forms, Schwarzian conditions, and symmetries of differential equations in physics
• The saga of the Ising susceptibility
• Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model
• Canonical decomposition of linear differential operators with selected differential Galois groups
• From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve
• On the diagonal susceptibility of the 2D Ising model
• Scaling functions in the square Ising model
• Renormalization, isogenies and rational symmetries of differential equations
• Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)
• Factorization of the Ising model form factors
• The lambda extensions of the Ising correlation functions C(M, N)
• Proof of the Borwein-Broadhurst conjecture for a dilogarithmic integral arising in quantum field theory
• On the Singularities in the Susceptibility Expansion for the Two-Dimensional Ising Model